proc imeche part c: analysis of transient …...analysis of transient amplification for a torsional...

Original Article

Analysis of transient amplificationfor a torsional system passingthrough resonance

Laihang Li and Rajendra Singh

Abstract

The classical problem of vibration amplification of a linear torsional oscillator excited by an instantaneous sinusoidal

torque is re-examined with focus on the development of new analytical solutions of the transient envelopes. First, a new

analytical method in the instantaneous frequency (or speed) domain is proposed to directly find the closed-form solu-

tions of transient displacement, velocity, and acceleration envelopes for passage through resonance during the run-up or

run-down process. The proposed closed-form solutions are then successfully verified by comparing them with numerical

predictions and limited analytical solutions as available in prior literature. Second, improved analytical approximations of

maximum amplification and corresponding peak frequency are found, which are also verified by comparing them with

prior analytical or empirical formulas. In addition, applicability of the proposed analytical solution is clarified, and their

error bounds are identified. Finally, the utility of analytical solutions and approximations is demonstrated by application

to the start-up process of a multi-degree-of-freedom vehicle driveline system.

Keywords

Analytical methods, instantaneous frequency domain analysis, non-stationary process, transient vibration, vehicle drive-

line system, rotor dynamics

Date received: 21 May 2014; accepted: 13 October 2014

Introduction

The transient amplification of a rotating systempassing through a resonance during speed run-up(or run-down) events is a classical vibration prob-lem that has been studied by many authors in thelast eight decades.1–20 It may be easily describedby the following equation for a simplified damped,linear time-invariant torsional oscillator where �� tð Þis the torsional displacement, �!2

1 is the nat-ural frequency, � is the damping ratio, and thebar over a symbol indicates a dimensionalparameter

�€� �t� �þ 2� �!1

�_� �t� �þ �!2

1�� t� �¼ �Te �t

� �ð1aÞ

�Te �t� �¼ �!2

1 �� sin ��0 �tþ1

2�� t2 þ �’

� �ð1bÞ

Here, �Te �t� �

is the external alternating (assumed to besinusoidal) torque with constant amplitude �!2

1 �� andphase �’, but its frequency (speed) ��0 þ ��t is assumedto vary linearly with a constant rate �� (rad/s2) asshown above; note that the mean torque and displace-ment terms usually drop out for a simplified linearsystem. The run-up and run-down speed events in

many rotating mechanical systems could be repre-sented by setting ��4 0 and ��5 0, respectively. Forexample, in the torsional driveline system of a groundvehicle, �Te �t

� �is from the engine start-up process,20

and ��0 þ �� t is related to the instantaneous firing fre-quency. Likewise, in a vehicle braking system, �Te �t

� �is

generated due to friction-controlled run-down pro-cess16–19 in which ��0 þ �� t is related to the wheelspeed. Newland15 has claimed that no analytical solu-tion of the transient envelope (amplification throughthe critical speed) exists. To overcome this void, a newinstantaneous frequency domain method will be pro-posed in this article, and the goal is to directly find theclosed-form solutions of the transient envelopes gen-erated by equation (1); only constant frequency(speed) acceleration rates for run-up and run-

Acoustics and Dynamics Laboratory, NSF Smart Vehicle Concepts

Center Department of Mechanical and Aerospace Engineering, The

Ohio State University, Columbus, OH, USA

Corresponding author:

Rajendra Singh, Acoustics and Dynamics Laboratory, NSF Smart Vehicle

Concepts Center Department of Mechanical and Aerospace

Engineering, The Ohio State University, 201 West 19th Avenue,

Columbus, OH 43210, USA.

Email: [email protected]

Proc IMechE Part C:

J Mechanical Engineering Science

2015, Vol. 229(13) 2341–2354

! IMechE 2014

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DOI: 10.1177/0954406214558148

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down processes are considered to ensure tractablesolutions.

Problem formulation

Three methods have been historically employed tofind the transient envelope of vibration displacementresponses in the context of an oscillator. First, thenumerical integration technique has been extensivelyused by many researchers including Lewis1 and Hok2

to solve the differential equation regarding theresponse amplitude. Second, a signal processing con-cept, such as the Hilbert transform method,21 hasbeen utilized by Sen et al.18,19 and Li and Singh20

to estimate the transient envelope from numericaland experimental responses. Third, a closed-formsolution of vibration displacement of a linear oscil-lator has been sought first, and then the transientenvelope is indirectly found in the time domain. Inparticular, Fearn and Millsaps3 found the analyticalsolution of the displacement of an undamped oscil-lator during the acceleration process, and Markertand Seidler12 and Sen et al.18,19 derived the analyticaltransient displacement envelope of a damped linearoscillator by using this indirect method. Thecommon theme with the above three methods isthat the transient envelope is estimated from thevibration response in time domain by using an indir-ect method.

Several prior researchers focused on the max-imum amplification level and the correspondingexcitation or peak frequency.1,3,12 A relationshipbetween the peak frequency and the critical speed(natural frequency) has only been qualitativelydescribed by Newland15 by using a computationalmethod. Some approximations or empirical for-mulas have been developed, based on numericalsolutions, to estimate the amplification levels.1,3,12

For example, Markert and Seidler12 proposed ananalytical approximation for displacement amplifica-tion and compared with empirical formulas; how-ever, analytical approximations of vibrationvelocity and acceleration’s amplification levels werenot found in Markert and Seidler.12 Therefore, thisarticle intends to provide refined analytical approxi-mations and a more quantitative description ofthe displacement, velocity, and accelerationamplifications.

The vibration amplification is essentially a non-stationary process as it is induced by an instantaneousexcitation frequency passing through the criticalspeeds or natural frequencies. Thus, the time (�t) andinstantaneous frequency ( ��0 þ �� t) domains arerelated by the constant acceleration rate ( ��) in equa-tion (1). Since the transient vibration depends on theinteraction between ��0 þ �� t and the natural fre-quency ( �!1), an instantaneous frequency domain isdefined as ��0 þ �� t

� �= �!1. The dimensionless form of

equation (1) is formulated as follows to facilitate fur-ther development

t ¼ �!1 �t, � ¼��

��, � ¼

��

�!21

, �0 ¼��0

�!1, � ¼

��

��,

!1 ¼�!1

�!1, ’ ¼ �’, � ¼

��0 þ ��t

�!1

ð2Þ

�� t� �¼ �� tð Þ, �_� �t

� �¼

�_� tð Þdt

d �t¼

�_� tð Þ �!1,

�€� �t� �¼

�€� tð Þdt

d�t

� �2

¼�€� tð Þ �!2

1

ð3Þ

€� tð Þ þ 2�!1_� tð Þ þ !2

1� tð Þ ¼ !21� sin �0tþ

1

2�t2 þ ’

� �ð4Þ

Specific objectives are as follows: (1) Develop a newanalytical method in the instantaneous frequency (orspeed) domain that would directly yield the closed-form solutions of transient envelopes of � tð Þ, _� tð Þ,and €� tð Þ and verify these expressions by comparingresults with those from the numerical integration ofequation (4); (2) Find and verify refined analyticalapproximations of the peak frequency and corres-ponding maximum amplifications (in the instantan-eous frequency or speed domain based on theanalytical solutions of the transient envelope) andcompare results with prior literature12; and (3)Demonstrate the utility of analytical solutions andapproximations by an application to the start-upevent of vehicle engine subsystem.20,22,23

Closed-form solutions oftransient envelopes

The closed-form solutions of equation (4) are soughtin � ¼ ��0 þ �� t

� �= �!1 domain. The sinusoidal torque

term is replaced below by a complex-valued torqueterm !2

1~�e j �0þ

�2tð Þt where j is the imaginary unity

and select ’ ¼ 0 without losing any generality. Thetorsional displacement � tð Þ is assumed as � tð Þ ¼~� �ð Þe j �0þ

�2ð Þt, where the complex-valued amplitude ~�

depends on a new variable � ¼ �t. Substitution ofthese two complex-valued expressions into equa-tion (4) yields the following equations, whered�dt t ¼ �t ¼ �

_�ðtÞ ¼d ~�ð�Þ

d�

d�

dte j �0þ

�2ð Þt þ ~�ð�Þ j

� �0 þ�

2

� �þ

1

2

d�

dt

� �t

� e j �0þ

�2ð Þt ð5Þ

€�ðtÞ ¼d2 ~�ð�Þ

d�2d�

dt

� �2

e j �0þ�2ð Þt þ 2

d ~�ð�Þ

d�

d�

dtj

2342 Proc IMechE Part C: J Mechanical Engineering Science 229(13)



� �0 þ�

2

� �þ

1

2

d�

dt

� �t

� e j �0þ

�2ð Þt þ ~�ð�Þ j

�d�

dtþ j �0 þ

�

2

� �þ

1

2

d�

dt

� �t

� �2 !

e j �0þ�2ð Þt ð6Þ

€~�ð�Þ�2 þ_~�ð�Þ 2�!1�þ j2� �0 þ �ð Þ½ �

þ ~�ð�Þ !21 � �0 þ �ð Þ

2þ j �þ 2�!1 �0 þ �ð Þð Þ

�¼ ~�!2

1 ð7Þ

Here, the instantaneous frequency domain � ¼�0 þ �t ¼ �0 þ � is incorporated into equation (7)as follows, where ~� �ð Þ ¼ ~� �ð Þ,

_~� �ð Þ ¼_~� �ð Þ,

€~� �ð Þ ¼€~� �ð Þ

€~� �ð Þ þ_~� �ð Þ

j2

��þ

2�!1

�

� �þ ~� �ð Þ

� �1

�2�2 þ

2�!1j

�2�þ

!21 þ j�

�2

� �¼ ~�

!21

�2ð8Þ

Therefore, equation (8) is solved to find the complex-valued amplitude ~� �ð Þ, instead of solving equation(4). First, the homogeneous solution of equation (8)is found as below

€~� �ð Þ þ_~� �ð Þ

j2

��þ

2�!1

�

� �þ ~� �ð Þ

� �1

�2�2 þ

2�!1j

�2�þ

!21 þ j�

�2

� �¼ 0

ð9Þ

€~� �ð Þ þ_~� �ð Þ A�þBð Þ þ ~� �ð Þ C�2þD�þE

� �¼ 0

ð10Þ

A ¼2j

�, B ¼

2�!1

�, C ¼ �

1

�2,

D ¼2�!1j

�2, E ¼

!21 þ j�

�2

ð11Þ

Next, ~� �ð Þ is transformed to ~� �ð Þ as follows, where sis an arbitrary constant

~� �ð Þ ¼ ~� �ð Þes�2

,_~� �ð Þ¼

_~� �ð Þes�2

þ2s� ~� �ð Þes�2

ð12a;bÞ

€~� �ð Þ ¼€~� �ð Þes�

2

þ 4s�_~� �ð Þes�

2

þ ~� �ð Þes�2

2sþ 4s2�2� �

ð13Þ

€~� �ð Þþ 4sþAð Þ�þBð Þ_~� �ð Þþ 2sþ 4s2þ2sAþC

� ��2þ 2sBþDð Þ�þE

�~� �ð Þ ¼ 0 ð14Þ

Equation (14) becomes a second-order differentialhomogeneous equation with respect to ~� �ð Þ. Inorder to eliminate the square term of � and to sim-plify equation (14), set 4s2 þ 2sAþ C ¼ 0, and

the values of s are calculated as s ¼ ð�A�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA2 � 4Cp

Þ=4 ¼ �j=2�. Then, substitute s back intoequation (14) to yield the following

4sþ A ¼ 0, 2sBþD ¼ 0 ð15a; bÞ

€~� �ð Þ þ B_~� �ð Þ þ 2sþ Eð Þ ~� �ð Þ ¼ 0 ð16Þ

The solutions of equation (16) are found below bysubstituting B and E where l1,2 ¼ !1=�ð Þ �

�� jffiffiffiffiffiffiffiffiffiffiffiffiffi1� �2

p� �€~� �ð Þ þ

2�!1

�_~� �ð Þ þ

!21

�2~� �ð Þ ¼ 0 ð17Þ

~�h �ð Þ ¼ C1el1� þ C2e

l2� ð18Þ

Then, the homogeneous solution of equation (10) iseasily obtained where C1 and C2 are arbitrary con-stants determined by the initial conditions

~�h �ð Þ ¼ ~C1el1� þ ~C2e

l2��

es�2

¼ C1el1�þs�2

þ C2el2�þs�2

ð19Þ

The particular solution of equation (8) can be foundbased on the homogeneous solution ~�h �ð Þ as below;here, W is Wronskian determinant24; the primesymbol describes the first derivative with respect to�, and erf (x) is the error function25

~�p �ð Þ ¼ el2�þs�2

Zel1�þs�2

Wd�

"

�el1�þs�2

Zel2�þs�

2

Wd�

#~�!2

1

�2ð20Þ

W ¼ el1�þs�2

el2�þs�2� �0

��el2�þs�2

el1�þs�2

� �0�

¼ e l1þl2ð Þ�þ2s�2

l2 � l1ð Þ ð21Þ

~�p �ð Þ ¼

ffiffiffi�p

2 l2� l1ð Þffiffisp

~�!21

�2es�

2þl2�þl224s erf

�

�l2þ 2s�

2ffiffisp

� �� es�

2þl1�þl214s erf

l1þ 2s�

2ffiffisp

� �ð22Þ

Now, ~� �ð Þ is found below by summing the homoge-neous solution ~�h �ð Þ and particular solution ~�p �ð Þwhere �5 1

~� �ð Þ ¼ ~�h �ð Þ þ ~�p �ð Þ

¼ C1el1�þs�2

þ C2el2�þs�2

þ

ffiffiffi�p

2 l2 � l1ð Þffiffisp

�~�!2

1

�2

es�2þl2�þ

l224s erf l2þ2s�

2ffiffisp

� ��es�

2þl1�þl214s erf l1þ2s�

2ffiffisp

� �2664

3775 ð23Þ

Li and Singh 2343



Then, the transient envelope of torsional displacement� tð Þ in the instantaneous frequency domain Ed �ð Þ isdirectly found as

Ed �ð Þ ¼ ~� �ð Þ ð24Þ

The coefficients C1 and C2 are determined from the ini-tial conditions ~� �0ð Þ and

_~� �0ð Þ as

~� �0ð Þ ¼ ~�0 ¼ C1~�0 1 þ C2

~�0 2 þ ~�0 3 ð25Þ

~�0 1 ¼ el1�0þs�20 , ~�0 2 ¼ el2�0þs�

20 ð26a; bÞ

~�0 3 ¼

ffiffiffi�p

2 l2� l1ð Þffiffisp

~�!21

�2es�

20þl2�0þ

l224s erf

�

�l2þ 2s�0

2ffiffisp

� �� es�

20þl1�0þ

l214s erf

l1þ 2s�0

2ffiffisp

� �ð27Þ

_~� �0ð Þ ¼ ~�0 ¼ C1~�0 1 þ C2

~�0 2 þ ~�0 3 ð28Þ

~�0 1 ¼ l1 þ 2s�0ð Þel1�0þs�20 ,

~�0 2 ¼ l2 þ 2s�0ð Þel2�0þs�20

ð29Þ

~�0 3

¼

ffiffiffi�p

~�!21

2 l2� l1ð Þffiffisp�2

�

2s�0þ l2ð Þes�20þl2�0þ

l224s erf

l2þ 2s�0

2ffiffisp

� �

þ es�20þl2�0þ

l224s2ffiffisp

e�

l2þ2s�02ffisp

� �2

ffiffiffi�p

0BBBBB@

1CCCCCA

�

2s�0þ l1ð Þes�20þl1�0þ

l214s erf

l1þ 2s�0

2ffiffisp

� �

þ es�20þl1�0þ

l214s2ffiffisp

e�

l1þ2s�02ffisp

� �2

ffiffiffi�p

0BBBBB@

1CCCCCA

2666666666666666664

3777777777777777775

ð30Þ

C2 ¼

~�0 � ~�0 3 �~�0� ~�0 3

~�0 1

~�0 1

~�0 2 �~�0 2

~�0 1

~�0 1

,

C1 ¼~�0 � ~�0 3 � C2

~�0 2

~�0 1

ð31a; bÞ

The transient envelopes, Ev �ð Þ and Ea �ð Þ of _� tð Þ and€� tð Þ, can be easily derived since ~� �ð Þ has been found.First, Ev �ð Þ is found as


d�

d�

dte j �0þ

�2ð Þt

þ ~�ð�Þ j �0 þ�

2

� �þ

1

2

d�

dt

� �t

� e j �0þ

�2ð Þt ð32Þ


d��e j �0þ

�t2ð Þt þ j ~�ð�Þ�e j �0þ

�t2ð Þt

¼d ~�ð�Þ

d��þ j ~�ð�Þ�

!e j �0þ

�t2ð Þt ð33Þ

Ev �ð Þ ¼_~�ð�Þ�þ j ~�ð�Þ� ð34Þ

Here,_~� �ð Þ ¼ C1

~�1 þ C2~�2 þ ~�3 is determined as

below

~�1 ¼ l1 þ 2s�ð Þel1�þs�2

, ~�2 ¼ l2 þ 2s�ð Þel2�þs�2

ð35a; bÞ

~�3

¼

ffiffiffi�p

2 l2 � l1ð Þffiffisp

~�!21

�2

�

2s�þ l2ð Þes�2þl2�þ

l224s erf

l2 þ 2s�

2ffiffisp

� �

þ es�2þl2�þ

l224s2ffiffisp

e�

l2þ2s�2ffisp

� �2

ffiffiffi�p

0BBBBBB@

1CCCCCCA

�

2s�þ l1ð Þes�2þl1�þ

l214s erf

l1 þ 2s�

2ffiffisp

� �

þ es�2þl1�þ

l214s2ffiffisp

e�

l1þ2s�2ffisp

� �2

ffiffiffi�p

0BBBBBB@

1CCCCCCA

266666666666666666664

377777777777777777775

ð36Þ

Similarly, Ea �ð Þ can be found as below where€~�ð�Þ�2

is neglected since the presence of �2 makes it negligiblecompared with other two terms

€�ðtÞ ¼€~�ð�Þ�2e j �0þ

�t2ð Þt þ 2

_~�ð�Þ�j�e j �0þ�t2ð Þt

þ j ~�ð�Þ �þ j�2� �

e j �0þ�t2ð Þt

ffi 2j_~�ð�Þ��þ j ~�ð�Þ �þ j�2

� �� e j �0þ

�t2ð Þt ð37Þ

Ea �ð Þ ffi 2j_~�ð�Þ��þ j ~� �ð Þ �þ j�2

� � ð38Þ

Verification of the closed-form solutionsfor transient envelopes

Two methods are utilized to verify the proposedclosed-form solutions. First, equation (4) is numeric-ally integrated by MATLAB26 to solve the transientresponses � tð Þ, _� tð Þ, and €� tð Þ, and then the results arecompared with the closed-form solutions Ed �ð Þ,Ev �ð Þ, and Ea �ð Þ. Since the initial conditions ( ~�0

and ~�0) of the proposed closed-form solutions arein the instantaneous frequency domain, a relationship




between the initial conditions (at t¼ 0) in the fre-quency and time domains is given as below

� tð Þ ¼ Im ~� �ð Þe j �0þ�2ð Þt

� �!

� 0ð Þ ¼ �0 ¼ Im ~� �0ð Þ� �

¼ Im ~�0

� �ð39Þ

_�ðtÞ ¼ Imd ~�ð�Þ

d��þ j ~�ð�Þ�

!e j �0þ

�t2ð Þt

!!

_� 0ð Þ ¼ �0 ¼ Im_~� �0ð Þ�þ j ~�ð�0Þ�0

� �¼ Im ~�0�þ j ~�0�0

� �ð40Þ

The following parameters and initial conditions areassumed to simplify the comparison for run-up cases(� ¼ 0:001 and 0.01 with �0 ¼ 0:7) and run-downcases (� ¼ �0:001 and �0.01 with �0 ¼ 1:5):� ¼ 1, � ¼ 10�6, �0 ¼ 0, �0 ¼ 0, ~�0 ¼ 0, ~�0 ¼ 0:Only a lightly damped system (� ¼ 10�6) is considereddue to the limitation of the error function calculationalgorithm used.25 Comparisons of Figures 1 and 2indicate that the proposed closed solutions (Ed �ð Þ,

Ev �ð Þ, and Ea �ð Þ) accurately describe the envelopesof the numerically obtained responses (� tð Þ, _� tð Þ, and€� tð Þ) for both run-up and run-down cases.

Further, Ed �ð Þ predictions from equations (23) and(24) in the instantaneous frequency domain are com-pared with the comparable closed-form solution oftransient envelope as proposed by Markert andSeidler.12 The detailed time domain derivation � tð Þof � tð Þ, by using the convolution theorem, can befound in Markert and Pfutzner’s6 paper, thoughthey provided only the displacement envelope. Byusing the same parameters, Ed �ð Þ and � tð Þ are com-pared in Figures 3 and 4 for run-up and run-downprocesses, respectively. For the sake of convenience,�ðtÞ is also plotted over � by scaling the t vector to the�0 þ �t vector without changing the nature of �ðtÞ.Since good agreement is seen in Figures 3 and 4, it isclear that the proposed direct analytical method, inthe instantaneous frequency domain, is accurate.

Figure 1. Verification of closed-form solutions for transient

envelopes as a function of instantaneous excitation frequency

during the run-up process. (a) � ¼ 0:001; (b) � ¼ 0:01.

Key: , closed-form solutions (Ed �ð Þ of � tð Þ, Ev �ð Þ of _� tð Þ,

Ea �ð Þ of €� tð Þ); , numerical predictions of system

responses � tð Þ, _� tð Þ, and €� tð Þ.

Figure 2. Verification of closed-form solutions for transient

envelopes as a function of instantaneous excitation frequency

during the run-down process. (a) � ¼ �0:001; (b) � ¼ �0:01.

Key: , closed-form solutions (Ed �ð Þ of � tð Þ, Ev �ð Þ of _� tð Þ,

Ea �ð Þ of €� tð Þ); , numerical predictions of system

responses (� tð Þ, _� tð Þ, €� tð Þ).

Li and Singh 2345



New analytical approximations of peakfrequencies and maximum amplifications

During a typical run-up or run-down process, themaximum amplifications of the transient envelopes(Ed max, Ev max, and Ea max) occur at three peak fre-quencies as observed in Figures 1, 2, and 3. Since suchpeak frequencies are very close to each other, only thepeak frequency of Ed �ð Þ is of interest and is denotedby �p in Figure 3. A new analytical method is pro-posed to estimate �p from the closed-form solution ofEd �ð Þ first, and then corresponding maximum ampli-fications Ed_max, Ev_max, and Ea_max are found. Therun-up process is examined first as an example casein the following derivation.

The effect of the damping ratio � on �p and Ed max

is numerically examined by using � ¼ 1, ’ ¼ 0,~�0 ¼ 0, ~�0 ¼ 0. The results of Figure 5 indicatethat Ed max, Ev max, and Ea max are significantlyaffected by �; however, � has a negligible effect on�p. Thus, it is reasonable to assume an undampedsystem (�¼ 0) for estimating �p, Ed max, Ev max, andEa max. Furthermore, the effect of the initial condi-tions on the amplification are not considered andj ~�0j � Ed max, ~�0

¼ 0.Next, Ed �ð Þ is plotted in Figure 6 along with

j ~�h �ð Þj, j ~�p �ð Þj to assess the roles of various partsin determining the amplification trend and peak fre-quency �p at � ¼ 0:01. Beyond the critical speed(�c ¼ !1¼ 1), Ed �ð Þ follows j ~�p �ð Þj; it indicates

Figure 3. Comparison between Markert and Seidler’s12 solution (denoted by � tð Þ) and proposed closed-form solution (denoted by

Ed �ð Þ) during the run-up process. (a) � ¼ 0:001; (b) � ¼ 0:01. Key: , � tð Þ; , Ed �ð Þ.

Figure 4. Comparison between Markert and Seidler’s12 solution (denoted by � tð Þ) and proposed closed-form solution (denoted by

Ed �ð Þ) during the run-down process. (a) � ¼ �0:001; (b) � ¼ �0:01. Key: , � tð Þ; , Ed �ð Þ.




that j ~�p �ð Þj should be analyzed in detail since itwould determine the amplification trend (�p andEd max). Divide ~�p �ð Þ into two parts ~�p1 �ð Þ and~�p2 �ð Þ

~�p1 �ð Þ ¼

ffiffiffi�p

~�!1

2ffiffiffiffiffiffiffi2�jp e

�j 12��

2þ!1� �þ

!21

2�

� �

� erf�þ !1ð Þ

2ffiffiffi�p j� 1ð Þ

� �ð41Þ

~�p2 �ð Þ ¼�

ffiffiffi�p

~�!1

2ffiffiffiffiffiffiffi2�jp e

�j 12��

2�!1� �þ

!21

2�

� �

� erf�� !1ð Þ


� �ð42Þ

Now, j ~�p1 �ð Þj and j ~�p2 �ð Þj are plotted with Ed �ð Þto examine the role of these two terms in determin-ing the amplification and �p. Figure 7 suggests thatj ~�p2 �ð Þj determines the peak frequency �p beyond�c ¼ 1. Therefore, ~�p2 �ð Þ should be examined fur-ther to estimate �p. In ~�p2 �ð Þ, the �

ffiffiffi�p

~�!1=�

2ffiffiffiffiffiffiffi2�jpÞe�jð

12��

2�!1� �þ

!21

2�Þ expression does not affect theamplification since this is an oscillatory term withan amplitude

ffiffiffi�p

~�!1=2ffiffiffiffiffiffiffi2�jp . Therefore, attention

must be directed to the error function erf �� !1ð Þ=ðð

2ffiffiffi�pÞ j� 1ð ÞÞ with a complex-valued argument

�� !1ð Þ=2ffiffiffi�p� �

j� 1ð Þ� �

. Unlike the continued frac-tion method used by Markert and Seidler,12 aninfinite series approximation25 is employed in thisarticle to analyze the error function with a com-plex-valued argument based on the following

Figure 5. Effect of damping ratio on transient envelope Ed �ð Þ as a function of instantaneous excitation frequency. (a) � ¼ 0:001;

(b) � ¼ 0:01. Key: , � ¼ 10�6; , � ¼ 0:01; , � ¼ 0:05.

Figure 6. Components of transient envelope Edð�Þ as a

function of instantaneous excitation frequency. Key: ,

Edð�Þ; , j ~�hð�Þj; , j ~�pð�Þj.

Figure 7. Comparison between estimated Edð�Þ

and two parts of j ~�pð�Þj. Key: , Edð�Þ;

j ~�p1ð�Þj; , j ~�p2ð�Þj.

Li and Singh 2347



where Z ¼ Rþ Ij is a complex-valued argument

erf Zð Þ ¼ erf Rð Þ þe�R

2

�

(1� e�2jRI

2Rþ 2

X1n¼1

e�n2=4

n2 þ 4R2

� 2R� e�2jRI 2R cosh nIð Þ � jn sinh nIð Þð Þ� �)

ð43Þ

Insert Z ¼ ��!1ð Þ

2ffiffi�p j� 1ð Þ into equation (43) and rewrite

this into four parts as shown below

erf�� !1ð Þ


� �

¼ erf �� !1ð Þ

2ffiffiffi�p

� �þ1 �ð Þ þ2 �ð Þ þ3 �ð Þ

ð44Þ

1 �ð Þ ¼

e�

��!12ffiffi�p

� �2

e2j

��!12ffiffi�p

� �2

� 1

0@

1A ffiffiffi

�p

� �� !1ð Þ,

2 �ð Þ ¼e�

��!12ffiffi�p

� �2

� �� !1ð Þð Þ

2ffiffiffi�p

�

X1n¼1

e�n2

4

n2

4 þ��!1

2ffiffi�p

� �2ð45a; bÞ

3 �ð Þ ¼ �e�

��!12ffiffi�p

� �2

þ2j��!12ffiffi�p

� �2

2�

X1n¼1

e�n2

4

n2

4 þ��!1

2ffiffi�p

� �2

�en

��!12ffiffi�p

2�

�� !1ffiffiffi�p � nj

� �

þe�n

��!12ffiffi�p

2�

�� !1ffiffiffi�p þ nj

� �!ð46Þ

Note that the erf � �� !1ð Þ=2ffiffiffi�p� �

term is boundedbetween �1 and 1. In addition, among 1 �ð Þ,2 �ð Þ, and 3 �ð Þ, one term is found from 3 �ð Þthat contributes most to the amplification as givenbelow

r �ð Þ ¼e�

��!12ffiffi�p

� �2

2�

X1n¼1

e�n2

4

n2

4 þ��!1

2ffiffi�p

� �2 en

��!12ffiffi�p

2

�� !1ffiffiffi�p

!

ð47Þ

At each n, the following is gained by introducing anew variable ¼ ��!1

2ffiffi�p

rn ð Þ ¼e� ð Þ

2

2�

e�n2

4

n2

4 þ ð Þ2en ð48Þ

For the run-up case when 4 0 or �4!1, the localpeak points for each rn ð Þ term can be found by set-ting _rn ð Þ ¼ 0 as follows

_rn ð Þ ¼

e n�ð Þ�n2

4 n2

4 n� 22 þ 1� ��

þ2 n� 22 � 1� ��

( )

2� n2

4 þ ð Þ2� �2 ¼ 0 ð49Þ

�24 þ n3 � 1þn2

2

� �2 þ

n3

4þ

n2

4¼ 0 ð50Þ

Note that �� is real and positive, and since �4!1 is ofinterest, �� !1ð Þ= 2

ffiffiffi�p� �

should be real and positive.As shown in Figure 7, j ~�p2 �ð Þj is amplified when 4 0;therefore, the root r n (which is real and greater thanzero) should determine �p. Different values of n areexamined next so as to find the most dominant n thatwould determine �p for �¼ 0.001 and 0.01. The firstfive values of n (n ¼ 1, 2, 3, 4, 5) are selected to findthe real and positive root r n, and the corresponding�p n ¼ !1þ 2

ffiffiffi�p

r n values are listed in Table 1. Basedupon the numerical prediction of Figure 1, �p is foundto be 1.067 for � ¼ 0:001 and 1.213 for � ¼ 0:01.Therefore, according to Table 1, n ¼ 2 is the dominantterm that would determine �p. Accordingly, �p is esti-mated as �p ¼ !1 þ 2

ffiffiffi�p

r 2 ¼ !1 þ 2ffiffiffi�p

for the run-up process. Conversely, for the run-down process, theerror function from ~�p2 �ð Þ becomes erf �� !1ð Þ=ðð

2ffiffiffiffiffiffiffi��pÞ jþ 1ð ÞÞ, and the new 3 �ð Þ term is now

3 �ð Þ ¼ �e�

��!12ffiffiffiffi��p

� �2

�2j��!12ffiffiffiffi��p

� �2

2�

X1n¼1

e�n2

4

n2

4 þ��!1

2ffiffiffiffiffi��p

� �2

�en


2

�� !1ffiffiffiffiffiffiffi��p � nj

� �

þe�n


2

�� !1ffiffiffiffiffiffiffi��p þ nj

� �!ð51Þ

Also, rn ð Þ becomes e� ð Þ2

2�e�n

2

4

n2=4þ ð Þ2e�n when is

changed to �� !1ð Þ= 2ffiffiffiffiffiffiffi��p� �

. Here, 5 0 is of

interest since �p has been found to be less than !1

during the run-down process.15–19 By following theprocedures of equations (49) to (50), the negative andreal root r n and the corresponding �p n ¼ !1þ

2ffiffiffiffiffiffiffi��p

r n can be found at each n. It turns out that

Table 1. The real and positive roots of equation (50) and its

corresponding �p n for the first five n terms.

n 1 2 3 4 5

r n 0.5 1 1.5 2 2.5

�p n (� ¼ 0:001) 1.032 1.063 1.094 1.126 1.158

�p n (� ¼ 0:01) 1.100 1.200 1.300 1.400 1.500




n¼ 2 is still the dominant term, and the corresponding�p is found as below

�p ¼ !1 � 2ffiffiffiffiffiffiffi��p

ð52Þ

The peak frequency �p for both run-up and run-downprocesses can be represented by the following com-bined expression where sgn (x) is a sign function

�p ¼ !1 þ 2sgn �ð Þffiffiffiffiffiffi�j j

pð53Þ

In order to verify equation (53), �p is calculated twoways. First, �p is numerically estimated from themaximum value of the Ed �ð Þ using MATLAB,26

and then �p is estimated by using equation (53). Asshown in Figure 8, the proposed analytical approxi-mation of �p accurately predicts the peak frequency.Further, 2sgn �ð Þ

ffiffiffiffiffiffi�j j

pin equation (53) indicates a shift

of the peak frequency �p from the critical speed(�c ¼ !1¼ 1) which depends on the value and direc-tion of �. This finding is consistent with the qualita-tive discussion reported in Newland’s15 book.

Regarding the maximum amplification Ed max at�p for an undamped system, Figures 6 and 7 are uti-lized again. By comparing the contributions of twoparts of ~�p �ð Þ, it is seen that ~�p1 �ð Þ leads to an aver-aged level (DC component) since j ~�p1 �ð Þj is relativelyflat over the whole � region, and ~�p2 �ð Þ determinesthe amplification trend. Thus, both contributionsmust be combined. For the run-up process, substitu-tion of �p ¼ !1 þ 2

ffiffiffi�p

into equation (42) leads to thefollowing ~�p2 �ð Þ expression, where e�2jerf j� 1ð Þ ffi

0:72þ 1:12j

~�p2 �p

� �¼�

ffiffiffi�p

~�!1

2ffiffiffiffiffiffiffi2�jp e�2jerf j� 1ð Þ

ffi

ffiffiffi�p

~�!1

4ffiffiffi�p �1:84� 0:40jð Þ ð54Þ

For ~�p1 �ð Þ, it is clear thatffiffiffi�p

~�!1= 2ffiffiffiffiffiffiffi2�jp� �

contrib-utes most to the DC value; thus, only

ffiffiffi�p

~�!1=2ffiffiffiffiffiffiffi2�jp� �

is taken into account. Accordingly, Ed max

can be approximated as the following where~� ¼ �e j’ ¼ � ¼ 1 with ’¼ 0, and !1 ¼ 1

Ed max ¼ ~�p2 �p

� � þ ffiffiffi�p

~�!1

2ffiffiffiffiffiffiffi2�jp

¼

ffiffiffi�p

�

4ffiffiffi�p 1� jð Þ

þ

ffiffiffi�p

�

4ffiffiffi�p �1:84� 0:40jð Þ

ffi 1:46

1ffiffiffi�p ð55Þ

Likewise, for the run-down process, the substitutionof �p ¼ !1 � 2

ffiffiffiffiffiffiffi��p

into equation (54) yields thefollowing

~�p2 �p

� �¼�

ffiffiffi�p

~�!1

2ffiffiffiffiffiffiffi2�jp e2jerf jþ 1ð Þ

ffi

ffiffiffi�p

~�!1

4ffiffiffiffiffiffiffi��p �1:84þ 0:40jð Þ ð56Þ

Then, Ed max is found as

Ed max ¼ ~�p2 �p

� � þ ffiffiffi�p

~�!1

2ffiffiffiffiffiffiffi2�jp

¼

ffiffiffi�p

�

4ffiffiffiffiffiffiffi��p �1:84þ 0:40jð Þ

þ

ffiffiffi�p

�

4ffiffiffiffiffiffiffi��p �1� jð Þ

ffi 1:46

1ffiffiffiffiffiffiffi��p : ð57Þ

The maximum amplification Ed max for both run-upand run-down processes can be combined into oneexpression as

Ed max ffi 1:461ffiffiffiffiffiffi�j j

p ð58Þ

Equation (58) is compared with the exact maximumvalues of Ed �ð Þ for both run-up and run-down pro-cesses in Figure 9, and a reasonable agreement isachieved. Some discrepancies exist between Ed max

and the exact maximum values; one reason is that theexact maximum amplifications for run-up and run-down processes might not be identical with a highervalue of �j j though equation (58) assumes these to beidentical. However, Ed max between the run-up andrun-down processes might not be consistent as shownin Figure 10: when �j j is relatively small (such as�j j5 0:004), Ed max for the run-up process is slightlyhigher than the run-down process; when �j j isincreased up to 0.006, the Ed max value for the run-down process is higher. Thus, equation (58) is a rea-sonable approximation.

Further, Ev max and Ea max can be analyticallyapproximated by using Ed max from equation (58)based up the assumption that �p of Ed �ð Þ, Ev �ð Þ,and Ea �ð Þ are close to each other. Since j ~�ð�Þ� is

Figure 8. Verification of analytical approximation of �p as a

function of accelerate rate. Key: , numerical solution; ,

analytical approximation.

Li and Singh 2349



more dominant than_~�ð�Þ�, the following approxi-

mation can be found from equation (34)

Ev �ð Þ ffi j ~�ð�Þ� ! Ev max ffi Ed max�p

ffi 1:46

1ffiffiffiffiffiffi�j j

p 1þ 2sgn �ð Þffiffiffiffiffiffi�j j

p� �ð59Þ

Similarly, Ea max can be analytically found from equa-tion (38) as

Ea �ð Þ ffi j ~� �ð Þ j�2� � ! Ea max ffi Ed max�

2p

ffi 1:461ffiffiffiffiffiffi�j j

p 1þ 2sgn �ð Þffiffiffiffiffiffi�j j

p� �2: ð60Þ

Estimated Ev max and Ea max are compared inFigure 11 with the exact maximum values of Ev �ð Þand Ea �ð Þ. Approximations are reasonable for

different � values, and the discrepancies are due totwo reasons: first, the peak frequencies of Ev �ð Þ andEa �ð Þ should be around �p, though it might not bethe exact �p value; second, the approximations ofequations (59) and (60) introduce some errors.

Some empirical formulas have been reported in theliterature12 for an undamped single-degree-of-free-dom (SDOF) system with a force or torque excitation.Such formulas focus on �p and Ed max only and weremost likely found by intensive numerical studies andcurve fitting. Also, only one analysis-based approxi-mation, as suggested by Markert and Seidler12 usingthe continued fraction method, is available in the timedomain. Comparisons between analytical and empir-ical approximations for an undamped system (�¼ 0)are given in Table 2 and Figure 12. Further, Figure 12compares the proposed analytical approximation,Leul’s formulas,12 and the exact solutions. The pro-posed approximation provides mathematical inter-pretation with reasonable accuracy.

Applicability of analytical solution anderror bounds

The proposed analytical and numerical solutionsclearly show that the transient amplification level

Figure 11. Comparison between numerical solution and

analytical approximation of Ev max and Ea max as a function of

acceleration rate. (a) Ev max; (b) Ea max. Key: , numerical

solution; , analytical approximation.

Figure 9. Verification of analytical approximation of Ed max as

a function of accelerate rate. Key: , numerical solution; ,

analytical approximation.

Figure 10. Comparison between run-up and run-down

numerical solutions of Ed max as a function of absolute

acceleration rate. Key: , run-down process; , run-up

process.




depends on the value of �j j. For instance, Figures 9and 10 demonstrate that the amplification level con-siderably decreases as �j j goes up to 0.01. Therefore,in order to examine more severe amplification, smallervalues of �j j must be considered. Prior research-ers9,10,15 have suggested the range of � between�0.01 and 0.01 for rotating machines, which limits

the dimensional acceleration or deceleration rate( �j j) up to 100 r/min/s.

The accuracy of proposed analytical solutions isexamined next as �j j is increased to 0.1. The proposedanalytical solutions are able to predict Ed max and �p

with a reasonable accuracy in Figure 13. The devi-ation in �p when � decreases to �0.1 is caused by

Table 2. Comparison (for �p and Ed_max) between analytical approximations and empirical formulas (as reported in the literature)

for an undamped, linear single-degree-of-freedom system.

Approximation

method Author(s) �p Ed_max

Analytical Proposed in this article 1þ 2sgn �ð Þffiffiffiffiffiffi�j j

p1:46=


p

Analytical Markert and Seidler, 200112 1þ 2:157sgn �ð Þffiffiffiffiffiffi�j j

p1:486=


p

Empirical expressions

(as summarized by

Markert and

Seidler12)

Lewis, 19321,12 1= 0:68ffiffiffiffiffiffi�j j

p� �� 0:353sgn �ð Þ

Katz, 194712 1þ 2:178sgn �ð Þffiffiffiffiffiffi�j j

p

Zeller, 194912 1:25=ffiffiffiffiffiffi�j j

p

Fearn and Millsaps, 19673,12 1þ 2:15sgn �ð Þffiffiffiffiffiffi�j j

p1= 0:68


p� �� 0:25sgn �ð Þ þ 0:025


p

Markert, 198812 1þ 2:07sgn �ð Þffiffiffiffiffiffi�j j

p1:41=


p

Leul, 199412 1þ 2:222sgn �ð Þffiffiffiffiffiffi�j j

p1= 0:76� 0:01sgn �ð Þð Þ


p� �

Figure 12. Comparison between numerical solution, empir-

ical formula and analytical approximation of �p and Ed max as a

function of acceleration rate. (a) �p; (b) Ed max. Key: ,

numerical solution; , analytical approximation; , Leul’s

empirical formula.

Figure 13. Error analysis of the analytical solution as a func-

tion of the acceleration rate. (a) �p; (b) Ed max. Key: ,

numerical solution; , analytical solution.

Li and Singh 2351



the fact that the system might not be able to reach itsmaximum amplification before it stops (� ¼ 0) due toan extremely rapid deceleration rate (�4� 0:08).Similar accuracy is achieved in Ev max though notshown in this article. However, the accuracy dropsto some extent when �j j50:08 due to the post-proces-sing of analytical solution. Recall from equations (37)and (38), the term

€~� �ð Þ�2e j �0þ�t=2ð Þt is neglected forthe limit of the error function calculation given in€~� �ð Þ. This term with �2 does not significantly affectthe accuracy of Ea �ð Þ and Ea max as shown inFigures 1, 2, and 11 until �j j is 0.08 when the neg-lected term becomes more dominant.

In a ground vehicle driveline system, the start-upprocess of a multi-cylinder internal combustionengine20,22,23 occurs considered with an extremelyhigh acceleration rate. For instance, �� can be up to3000 r/min/s for a diesel engine, and the first engine-transmission subsystem vibration mode, �!1, is typic-ally around 15Hz or even higher. This leads to� ¼ ��= �!2

1 ffi 0:04, and thus, the practical limit of � isselected as �0:04, 0:04½ � in which the proposed ana-lytical solution is reasonably accurate.

Application to a vehicle driveline system

The start-up process of a multi-cylinder internal com-bustion engine20,22,23 is examined next. First, anundamped, linear torsional oscillator (�¼ 0) is con-sidered under which the most severe amplificationshould occur at a given �. Therefore, the followingpractical ranges are calculated with the practicallimit of � and � ¼ !1 ¼ 1 by using equations (53),(58), (59), and (60): for �p, ½0:6, 1:4�; for Ed_max,½7:3, þ1�. For the calculations of Ev max andEa max, run-up and run-down cases are separately con-sidered. For a run-up process, Ev_max: ½10:22, þ1�,Ea_max: ½14:31, þ1�; for a run-down process, Ev_max:½4:38, þ1�, Ea_max: ½2:63, þ1�.

The utility of the analytical solutions is furtherexplained in the context of an undamped, linear 4DOF, semi-definite torsional driveline subsystem ofa ground vehicle with a four-cylinder internal com-bustion engine. Here, �I1, �I2, �I3, �I4 represent the tor-sional inertias of the engine, the flywheel, theclutch, and the lumped transmission, respectively;�K12, �K23, �K34 represent the torsional stiffness of thecrankshaft, the clutch damper, and the input shaft,

respectively. The governing equations are describedin a dimensional form by �I�€� þ �K �� ¼ �T, where �T ¼

�Tm þ �Ta �t� �

0 0 �Td

�Tand the inertia and stiff-

ness matrices are given as

�I ¼

�I1 0 0 0

0 �I2 0 0

0 0 �I3 0

0 0 0 �I4

2666664

3777775,

�K ¼

�K12 � �K12 0 0

� �K12�K12 þ �K23 � �K23 0

0 � �K23�K23 þ �K34 � �K34

0 0 � �K34�K34

2666664

3777775ð61a; bÞ

In the �T ¼ �Tm þ �Ta �t� �

0 0 �Td

�Texpression, the

drag torque �Td is from the transmission lubricant andthe mean torque �Tm is generated by the engine. Notethat �Tm ¼ �Td ¼ 0 in the start-up mode. Thus, thetransient response is excited by the alternating partof engine torque �Ta �t

� �which is expressed byP

n

�Ta n sin n ��0 �tþ 0:5 �� t2� �

þ �’n� �

, where ��0 þ �� t isthe instantaneous crankshaft speed.22,23 Since thesecond order (n¼ 2) is usually dominant for a four-cylinder engine, only the dominant torque term�Ta 2 sin 2 ��0 �tþ 0:5 ��t2

� �þ �’2

� �is examined with

�’2 ¼ 0. Like equation (2), the dimensionless form ofthe governing equations in matrix form is

I €� þ K� ¼ T,

T ¼ !21� sin 2 �0tþ

1

2�t2

� �� 0 0 0

� Tð62a; bÞ

I¼

�I1�

�I1þ �I2 0 0 0

0�I2�

�I1þ �I2 0 0

0 0�I3�

�I1þ �I2 0

0 0 0�I4�

�I1þ �I2

2666666664

3777777775ð63Þ

K ¼

�K12�

�I1 þ �I2� �

�!21

� �K12�

�I1 þ �I2� �

�!21

0 0

� �K12�

�I1 þ �I2� �

�!21

�K12 þ �K23�

�I1 þ �I2� �

�!21

� �K23�

�I1 þ �I2� �

�!21

0

0 � �K23�

�I1 þ �I2� �

�!21

�K23 þ �K34�

�I1 þ �I2� �

�!21

� �K34�

�I1 þ �I2� �

�!21

0 0 � �K34�

�I1 þ �I2� �

�!21

�K34�

�I1 þ �I2� �

�!21

2666666664

3777777775

ð64Þ




Here, �!1 is the dimensional natural frequency of thefirst vibration mode of an engine-transmission subsys-tem; it is controlled by the clutch damper stiffnessduring the engine start-up process.20 By focusing onthe relative torsional displacement (� tð Þ ¼�2 tð Þ � �3 tð Þ) between the flywheel and the clutch,equation (62) is approximated by the followingdimensionless governing equation of the correspond-ing SDOF torsional system, where

�K23ð �I1þ �I2þ �I3þ �I4Þ

ð �I1þ �I2Þð �I3þ �I4Þffi �!2

1

€�2 tð Þþ�K23

�I1þ �I2� �

�!21

�2��3ð Þ¼!21� sin 2 �0tþ

1

2�t2

� �� ,

�I3þ �I4�I1þ �I2

€�3 tð Þþ�K23

�I1þ �I2� �

�!21

�3��2ð Þ¼0 ð65a;bÞ

€� tð Þ þ�K23

�I1 þ �I2 þ �I3 þ �I4� �

�I1 þ �I2� �

�I3 þ �I4� �

�!21

� tð Þ

¼ !21� sin 2 �0tþ

1

2�t2

� �� ! €� tð Þ þ � tð Þ

ffi !21� sin 2 �0tþ

1

2�t2

� �� ð66Þ

The results of Figure 14 with � ¼ 0:0005 clearly showthat the closed-form solutions of the SDOF systemfrom equations (24), (34), and (38) accurately describethe transient envelopes of the 4 DOF semi-definitetorsional system in the instantaneous speed domain.Further, the amplifications at other orders(n¼ 0.5, 1, 1.5 , . . .) can be predicted as well based onthe principle of superposition. Finally, the amplifica-tion levels of the 4 DOF driveline system are approxi-mated by equations (53), (58), (59), and (60) andsummarized in Table 3 though now � and �p mustbe replaced by 2� and 0.5 �p as the n¼ 2 order in theinstantaneous speed domain is considered. The com-parison of Table 3 suggests that the SDOF systemmodel as given by equations (53), (58), (59), and(60) reasonably approximate the transient amplifica-tion of the 4 DOF torsional system.

Conclusion

Overall, it is satisfying to find mathematical solutionsthat have eluded many prior researchers. In particu-lar, the chief contributions of this article are as fol-lows. First, new closed-form solutions of the transientenvelopes of displacement, velocity, and accelerationamplifications are successfully developed in theinstantaneous frequency domain and verified withthe results from numerical predictions and prior lit-erature.6,12 Specifically, the closed-form solutions ofvelocity and acceleration envelopes are proposed forthe first time. Second, a new analytical approximationmethod is developed (based on the proposed closed-form solutions) to find the maximum amplificationsand corresponding peak frequencies which are verifiedby using the numerical (exact) solution. Comparisonswith prior empirical formulas1,3,12 and qualitativedescriptions15 verify proposed approximationswhile providing an analytical interpretation with

Figure 14. Comparison between the 4DOF versus SDOF

torsional systems for the transient envelopes as a function of

the instantaneous crankshaft speed during the run-up process

when second order torque is considered and �¼ 0.0005. Key:

, closed-form solutions for SDOF system (Ed �ð Þ of � tð Þ,

Ev �ð Þ of _� tð Þ, Ea �ð Þ of €� tð Þ); , numerical predictions of

the 4DOF system � tð Þ, _� tð Þ, and €� tð Þ.

Table 3. Comparison between the numerical solutions of the 4 DOF torsional system and analytical approximations based on the

linear SDOF torsional system.

�p Ed_max Ev_max Ea_max

Value of 2��p of 4 DOF

system

0.5 �p from

equation (53)

4 DOF

system

SDOF system

equation (58)

4 DOF

system

SDOF system

equation (59)

4 DOF

system

SDOF system

equation (60)

0.001 0.53 0.53 45.39 46.17 45.06 49.08 47.33 52.20

0.01 0.59 0.60 13.42 14.6 15.05 17.52 16.90 21.02

DOF: degree-of-freedom; SDOF: single-degree-of-freedom.

Li and Singh 2353



reasonable accuracy. In addition, the analyticalapproximations for maximum velocity and acceler-ation amplifications fill the gap in the prior literature.The applicability of analytical solutions is clarified aswell, and their error bounds are established. Third,the utility of the proposed analytical solutions is suc-cessfully demonstrated by the transient amplificationof a linear 4 DOF driveline system. The main limita-tion of this article is that the accuracy of velocity andacceleration approximations needs to be improvedsince calculation algorithms of first and secondderivatives of the error function with a complex-valued argument are error-prone.24–26

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest withrespect to the research, authorship, and/or publication ofthis article.

Funding

The author(s) disclosed receipt of the following financialsupport for the research, authorship, and/or publicationof this article: The Eaton Corporation Clutch Division

and the Smart Vehicle Concepts Center (www.SmartVehicleCenter.org) under the National ScienceFoundation Industry/University Cooperative ResearchCenters program (www.nsf.gov/eng/iip/iucrc) are acknowl-

edged for partially supporting this work. Individual contri-butions from L. Pereira, B. Franke, P. Kulkarni, J. Dreyer,and M. Krak are gratefully appreciated.

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